In this paper we present a result of existence of infinitely many arbitrarily small positive solutions to the following Dirichlet problem involving the p-Laplacian, where Ω ∈ RN is a bounded open set with sufficiently smooth boundary ∂Ω, p > 1, λ > 0, and f: Ω × R → R is a Carathéodory function satisfying the following condition: there exists t̄ > 0 such that Precisely, our result ensures the existence of a sequence of a.e. positive weak solutions to the above problem, converging to zero in L∞(Ω).

In this paper we deal with the existence of weak solutions for the following Neumann problemwhere ν is the outward unit normal to the boundary ∂Ω of the bounded open set Ω ⊂ R N . The existence of solutions, for the above problem, is proved by applying a critical point theorem recently obtained by B. Ricceri as a consequence of a more general variational principle.

Let Ebe a real separable and reflexive Banach space, Xc_ E weakly closed and unbounded, ,I and two non-constant weakly sequentially lower semicontinuous functionals defined on X, such that + A is coercive for each A > 0. In this setting, if inf sup((I)(x) + A((x) + p)) sup inf((x) + A((x) + p)) xX >_o A>O xX for every p E R, then, one has inf sup((x)+ A(x) + h(A)), sup, x_>o xexinf((x) + A(x) + h(A)) xex ,>_o for every concave function h'[O, +[ R.

We establish a multiplicity result to an eigenvalue problem related to second-order Hamiltonian systems. Under new assumptions, we prove the existence of an open interval of positive eigenvalues in which the problem admits three distinct periodic solutions

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